Aquarium filteration and heater systems are not required for aquarium tanks less than five gallons and in most instances should be left out altogether.
This is especially the case when it comes to Betta fish.
Generally speaking, the gravel that you place in your tank in most cases provides more than adequate biological filtration and again when it comes to Betta fish this is the case.
Everything should be alright for your Betta fish providing that you change its water regularly, and rid the tank of any excess food particles and other debris material as soon as possible.
Any tank or bowl size smaller than say a gallon, you will need to change all of the water once a day or at the very least three times per week. For much bigger tanks, you should do a portion of water about twenty five percent every week.
Make sure that you completely siphon and/or vacuum the container gravel each time you decide to change the water.
As you probably know already Bettas breathe really well at the surface of the tank water, and the water requires no form of aeration either. In some cases, aerating (and, circulating) the water could easily produce too much water flow for your pet Betta. Bettas are not used to swimming against a hard current or flow of water.
If you do decide to purchase a water heater for your aquarium tank, you should also purchase a thermometer too. Why? Because it will allow you to monitor the water temperature more accurately.
Keep in mind that extreme fluctuation of the water temperature can oftentimes kill or harm your Betta fish.